from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(432, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([18,27,26]))
pari: [g,chi] = znchar(Mod(227,432))
Basic properties
| Modulus: | \(432\) | |
| Conductor: | \(432\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(36\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 432.bj
\(\chi_{432}(11,\cdot)\) \(\chi_{432}(59,\cdot)\) \(\chi_{432}(83,\cdot)\) \(\chi_{432}(131,\cdot)\) \(\chi_{432}(155,\cdot)\) \(\chi_{432}(203,\cdot)\) \(\chi_{432}(227,\cdot)\) \(\chi_{432}(275,\cdot)\) \(\chi_{432}(299,\cdot)\) \(\chi_{432}(347,\cdot)\) \(\chi_{432}(371,\cdot)\) \(\chi_{432}(419,\cdot)\)
sage: chi.galois_orbit()
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Related number fields
| Field of values: | \(\Q(\zeta_{36})\) |
| Fixed field: | 36.36.5532004127928253705369187176396364210546696053048780432717505515499814912.1 |
Values on generators
\((271,325,353)\) → \((-1,-i,e\left(\frac{13}{18}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
| \( \chi_{ 432 }(227, a) \) | \(1\) | \(1\) | \(e\left(\frac{13}{36}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{23}{36}\right)\) | \(e\left(\frac{1}{36}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{35}{36}\right)\) | \(e\left(\frac{17}{18}\right)\) |
sage: chi.jacobi_sum(n)
Gauss sum
sage: chi.gauss_sum(a)
pari: znchargauss(g,chi,a)
Jacobi sum
sage: chi.jacobi_sum(n)
Kloosterman sum
sage: chi.kloosterman_sum(a,b)